Chapter 2: Scientific notation, symbols, units and equations

Scientific Notation

Scientific notation allows us to easily represent very big or very small numbers.

Some examples:The speed of light is approximately three hundred million metres per second. We write this number mathematically as follows: 300 000 000 m s-1 or, using scientific notation, 3 × 108 m s-1

It takes approximately 200 000 (2 × 105) Joules of heat to boil a kettle and 50 000 000 (5 × 107) Joules to heat a bath of water.

We can also use prefixes as shorthand for some scientific notation:

1 thousandth .001 1 × 10-3

1 millionth .000 001 1 × 10-6

1 billionth .000 000 001 1 × 10-9

1 thousand 1000 1 × 103

1 million 1000 000 1 × 106

1 billion 1000 000 000 1 × 109

For example 1 million Joules = 1 × 106 J = 1 Megajoule = 1 MJ.0052 metres = 5.2 × 10-3 m = 5.2 millimetres = 5.2 mm

See also the log tables on page 45

1

Prefix Symbol Factormilli- m × 10-3

micro- μ × 10-6

nano- n × 10-9

pico- p × 10-12

kilo- k × 103

mega- M × 106

giga- G × 109

tera- T × 1012

Try to identify the name or the term using the clues below

1. 1 x 1012 firmas

2. 2 x 1012 bulls

3. 1 x 109 lows

4. 2 x 106 phones

5. 1 x 103 manjaros

6. 1 x 103 whales

7. 2 x 103 mockingbirds

8. 1 x 10 -3 pedes

9. 1 x 10-3 nnium

10. 1 x 10-3 taries

11. 2 x 10-6 scopes

12. 3 x 10-6 phones

13. 1 x 10-12 boos

Answers1. 1 terra firma2. 1 terabull3. 1 gigalow4. 2 megaphones5. 1 kilomanjaro6. 1 kilowhale7. 2 kilomockingbird8. 1 millipede9. 1 millennium (so 1 nnium = 106 years)10. 1 military11. 2 microscopes12. 3 microphones13. 1 picaboo

Question: What is the unit for the level of beauty required to launch a single ship?

Answer: The milliHelen

2

3

SYMBOLS UNITS AND EQUATIONS

‘Maths is what you have left when you start with something interesting and take away the units.’

Well I still get full marks for a maths question if I don’t write down the formula?Yes, students will be awarded full marks for formula and for substitution if they only present the correctly substituted formula.However there is a much greater risk of making an error in substitution if the student hasn't the original formula written down and that results in zero marks.  This error is quite common.  Best practise: write down the formula!!

Note:All units are spelled out using lower case, e.g. newtons, joules, volts, kilogram.Symbols of units that derive from the name of a physicist are all uppercase e.g. J, V etc. while symbols for all other units remain lowercase, e.g. the symbol for the kilogram is kg.http://physics.nist.gov/cuu/Units/checklist.htmlhttp://physics.nist.gov/cuu/pdf/typefaces.pdf

(If typing these at any stage, note that both variables and constants should be italicised: v = u + at rather than v = u + at.)

Check that you know these by covering over all but the first column.

Let me know if I’ve missed any.

4

Mechanics

Quantity Symbol Unit Symbol Equation

Area a metres squared m2

Volume v metres cubed m3

Mass m kilogram kg

Density kilogram per metre cubed

kg m-3 = m/v

Displacement s metre m

Velocity v metre per second m s-1 v = d/t

Acceleration a metre per second squared

m s-2

Force F newton N F = ma

Momentum kg m s-1 = mv

Pressure p pascal Pa p = F/a

Moment of a force

newton metre N m

Torque (couple) T newton metre N m T = F x d

Energy E / Q / W joule J

Work w joule J W = F s

Power p watt W P = W/t

Angle (“theta”) radian rad

Angular velocity (“omega”) radian per second rad/sec = /t

5

Heat and Temperature

Quantity Symbol Unit Symbol Equation

Heat Capacity C joule per kelvin J/K Q = c ()

Specific Heat Capacity

c J/kg/K Q = mc

Latent Heat l joule per kilogram J/kg Q = ml

Waves, Sound and Light

Quantity Symbol Unit Symbol Equation

Frequency f hertz Hz

Wavelength (“lamda”) metres m

Velocity v (or c for light)

metre per second m/s v = f

Intensity I watts per metre squared W/m2 S.I. = P/A

Sound Intensity Level

decibels dB

6

Electricity

Quantity Symbol Unit Symbol Equation

Charge Q coulomb C

Electric Field Strength

E newtons per coulomb N/C E = F/Q

Potential Difference(“voltage”)

V volts V W = V Q

Capacitance C farads F C = Q/V

Current I amperes (amps) A I = Q/t

Power P watt W P = VI

Resistance R ohm Ω R = V/I

Resistivity ohm-metre Ω m = RA /l

Magnetic Flux Density

B tesla T F = BIL

Magnetic Flux Psi (“sigh”)

weber W = BA

Half-Life T1/2 second T1/2 = 0.693/

7

EQUATIONS

Many of the maths questions on the Leaving Cert Physics paper rely on you being able to quickly recall short equations.And yes these are all in the log tables, but if you are looking for an A or B grade then you don’t have time to go searching.The variables have deliberately not been arranged in the order in which they would appear in the formula (because that would just be too easy).

To test yourself, cover the third column and see if you can come up with the relevant equation given the information in the second column.

If you come across any equations which I have omitted, please let me know and I will update the list.

Hangman takes on a new dimension if you can include equations by allowing spaces for division, power s(e.g. ^2) etc.

Mechanics

Variables Equation

Equations of Motion v = u + ats = ut + ½ at2

v2 = u2 + 2as

Force, Mass and Momentum

acceleration, force, mass F = ma

weight , mass W = mg

velocity, mass, momentum = mv

Conservation of Momentum

m1 u1 + m2 u2 = m1 v3 + m2 v4

Pressure area, pressure, force P = F/A

density, height, pressure P = gh

Boyle’s Law P1V1= P2V2

Newton’s Law of Gravitation

gravitational force between two masses

Fg=Gm1 m2

d2

g at different heights acceleration due to gravity and distance above a planet

g = GM/ d2

Moment of a force distance, moment, force Moment = Force x distance

Torque force, distance, torque T = F x d (between forces)

Work, Energy force, work, displacement W = F s

Kinetic Energy velocity, mass energy Ek = ½ mv2

8

Potential Energy height, mass, energy Ep = mgh

Conservation of Energy mgh = ½ mv2

Power time, power work P = W/t

Percentage Efficiency Power Out / Power In x 100/1

Circular Motion time, angular velocity, theta = /t

linear velocity, angular velocity, radius

v = r

acceleration,angular velocity, radius, a = r2

linear velocity, radius, accelerationa = v2/r

force, angular velocity, radius, massF = mr2

mass, linear velocity, radius, force,F = mv2/r

mass of planet, acceleration due to gravity, radius of satellite

g = GM/R2

mass of a planet, radius,periodic tiime

T 2= 4 π 2R3

GM

Hooke’s Law extension, restoring force F = -k s

S.H.M. acceleration and displacement a = -2 s

periodic time and angular velocity T = 2/

frequency and periodic time T = 1/f

Simple Pendulum T = 2 l/g

9

Waves, Sound, Light

Mirrors image distance, magnification, Object distance M= vu

image height, magnification, object heightM=

imageHgt

object Hgt

image distance, magnification, object distance 1f= 1

u+ 1

v

Refraction Sini

Sinr=consant

real and apparent depth realdepth

apparentdepth=intersect

reversing direction and critical anglex intersect y= 1

y intersect x

refractive index and speeds C 1C 2

=intersect

refractive index and critical angleintersect = 1

SinCLenses image distance, mag,

object distance M= vu

image height, mag, object heightM=

imageHgt

object Hgt

image distance, magnification, object distance 1f= 1

u+ 1

vpower, focal length P=1

f

Addition of powers PTotal = P1 + P2

Waves Wavelength, velocity, frequency v = f

Doppler Effectf '= fc

c±u

Area, Power, Solar Intensity S.I. = Power / Area

Tension, Frequency, Lengthf = 1

2l √Tμ

Wavelength of light n = d Sin

Diffraction Grating Formula

Distance between slits on a diffraction grating d = 1/n

10

Electricity

Variables EquationStatic Electricity Coulomb’s Law

F =

14 πε

Q1 Q2

d2

Relative Permittivity = r o

Electric Field Intensity E = F/Q

Electric Field StrengthE = F =

Q4 πεd

2

Potential Difference Charge, Voltage, Work W = QV

Capacitance Charge, Potential difference, Capacitance C= Q/V

Area, Capacitance Distance C = A/d

Work/energy, Voltage Capacitance W = ½ CV2

Current, Charge, Time I = Q/t Q = It

Power, Current, Voltage P = VI

Ohm’s Law R=VI V = IR

Resistivity R = l/A

Wheatstone Bridge R1

R2=

R3

R4

Current, TimeEnergy, Resistance,

Heat = I2Rt

Joule’s Law Current, Power, Res Power = I2R

Current, Length,Force, Mag field density F = BIL

Force, Charge, velocity,Mag field density, F = Bqv

Magnetic Flux Density,Area, Magnetic Flux

= BA

Induced emf E = - N (d/dt)

Vrms, Maximum voltage Vrms= Vmax/(2)

Irms, Maximum current Irms = Imax/(2)

Transformer ViVo

=NpNs

11

Modern Physics

Variables Equation Year

Force on an electron mv2/r = Bev

Potential energy andKinetic energy of electron

eV = ½ mv2

Photoelectric Effect hf = + ½mv2

Frequency,Energy of a photon

E = hf

Wavelength,Energy of a photon

E = hc/

Decay rate,Decay constantNumber of atoms

dn/dt = N

Half life,Decay constant T1/2 = 0.693/

Energy,Mass

E = mc2

H11 + Li3

7 He2

4+He24

+ K.E.

Pair Production γ rays e- + e+ + K.E.

Particle Annihilation e- + e+ 2γ + K.E.

12

Advice from Physics teacher David Hobson

Be familiar with the log tables Pages 50 to 63 are the most important - they contain most of the formulas that you need.Not all the formulas are relevant - 

when revising a topic go through your copy and highlight the ones you need to know. Write notes about each formula, the context in which it is used, what the letters stand for and maybe

even an example of using it.The Prefixes used in SI units are on page 45.

The Fundamental Physical constants are given on pages 46 - 47.

For Physics use the Periodic Table on page 79 and the first table on page 82.

Many of the maths questions on the Leaving Cert Physics paper rely on you being able to quickly recall short equations.While most of these are available in the log tables, a good student shouldn’t need to look them up.

To test yourself cover the third column and see if you can come up with the relevant equation given the information in the second column

Hangman takes on a new dimension if you can include equations by allowing spaces for division, powers (e.g. ^2) etc.

The variables have deliberately not been arranged in the order in which they would appear in the formula (because that would just be too easy)

Those formulae which are highlighted are NOT in the log tables.See below for a list of formulas NOT in the log tables or in a different form to that in the log tables

13

MechanicsBoyle’s Law Volume of gas and Pressure pV = k OR p1V1= p2V2

Conservation of Energy Gravitational Potential Energy and Kinetic Energy mgh= 12 mv2

WeightGiven

g= Fm

Gravity & Circular Motion Velocity, radius of orbit and mass of central body v2=G MR

Components of a Vector Horizontal x=V cosθ and vertical y=V sin θ

Waves, Sound, LightMirrors & Lenses Magnification, Image height, Object height

(or size in any direction) m=Image HeightObject Height

Refraction Real and apparent depthn=Real Depth

Apparent DepthReversing direction and refractive indices

1 n 2=1

2 n1

Sound Intensity Sound Intensity, Area, Power Intensity, I = Power / Area

Dedibels Decibels and sound intensity Double I = an increase of 3 dB

Speed of sound Standing wave in tube closed at one end c= f [ 4 ( L+0 .3 d )]

Grating Formula Distance between slits on a diffraction grating d = 1/n

ElectricityStatic Electricity Relative Permittivity = r o

Electric Field Strength (Due to Q)E =

14 πε

Qd2

Current/Charge Current, Charge, Time Q = It OR I = Q/t

Joule’s Law Power, Current, Resistance Power=RI 2 (Heat=RI 2 t )

Magnetic Induction Induced E.M.F. in a coil with N turns E=−N d ( φ)dt

=d ( NBA )

dt

Transformer Power in = Power out V ¿ I ¿=V out I out

Modern Physics

Force on an electron Electron moving in a magnetic field moves in a circle Bev=mv2

r

Ek of an electron Kinetic energy of electron (V is voltage)

Half life Half-life, Decay constant T1/2 = 0.693/

Walton Split nucleus and release energy H11

+ Li3

7

He2

4+He24

+ K.E.

Pair Production Photon to particles (Note: one photon) γ photon e– + e+ + K.E.

14

Formulas not in tables or in a different form

Particle Annihilation Particles to photons (Note: two photons) e- + e+ 2γ photons + K.E.

15